EEET 3028 Communication Systems University of South Australia School of Engineering Assignment 2 (2017 SP2) General information This assignment covers sampling and principles of digital communication systems, and their simulation in MATLAB. Your report should include written answers, block diagrams, sketches of signals, plots produced by MATLAB and the corresponding MATLAB code. Explain how you came to your answers and conclusions, and provide concise answers. Hand-written answers, mathematical derivations, sketches, and block diagrams are fine if done neatly and clearly. You may scan them and include them in your document. The report is to be submitted online as one single PDF file and on time. Late submissions may not be considered. Make sure that you have added the cover page. The deadline for submission is 11pm Monday 11 June 2017. Problems Sampling 1. Consider the following signal x(t) = sinc2(10t): Write the Fourier transform X(f) of x(t). Sketch the magnitude spectrum. (2 marks) 2. What is the Nyquist rate fN for this signal? (1 marks) 3. The sampling function s(t) = +1 X i=−1 δ(t − iTs − T1); with sampling period Ts = 2f 1N , and T1 = T 5s is used to obtain the sampled signal y(t) = x(t) × s(t). Sketch the magnitude spectrum jY (f)j of y(t). Can you perfectly reconstruct the signal x(t) from y(t)? Explain why (or why not)? (5 marks) Note: You can do this without computing the spectrum Y (f) explicitly. 4. Repeat Question 3 with sampling period Ts = 0:91fN . (5 marks) 5. If perfect reconstruction is not possible in Question 3 or 4, what additional step can be employed prior to sampling to reduce distortion. Explain. (2 marks) Assignment 2 (2017 SP2) Page 1 of 4EEET 3028 Communication Systems University of South Australia School of Engineering 6. MATLAB simulation: reconstruct x(t) from the samples y. (a) Plot the signal x(t) in the range [-0.4:0.4] with time step T 5s . Obtained the samples at sampling time dictated in Question 3. On the same figure, stem plot the signal y in the range [-0.4:0.4]. (b) Let the reconstruction filter impulse response be h(t) = sinc( 2T t s ), t = [−6Ts : Ts=5 : 6Ts]. On MATLAB, the reconstruction can be implemented by • upsampling y by 5 to create yup. • convolving yup with h(t). Plot the reconstructed signal. Compare with the original signal and explain any discrepency. (5 marks) Digital transmitter b1; b2; : : : Mapping Pulse shaping p(τ) × A c cos 2πfct ak x(t) xc(t) 00 -3 01 -1 10 1 11 3 Mapping and constellation Symbol period: Ts = 2ms Pulse shape: p(τ) = sinc Tts rect  2T t s. Figure 1: Block diagram of a digital transmitter 7. Explain the roles of the blocks in Figure 1 (5 marks). 8. Calculate the average energy (Es) of the constellation. (2 marks) 9. What are the criteria (in both time and frequency domain) for intersymbol-interference (ISI) free transmission. Explain whether ISI-free is satisfied by the system in Figure 1. (6 marks) 10. Assume the data bit sequence is 011110001001. Compute ak; k = 1; 2; : : :. Compute and plot x(t) using MATLAB. (5 marks) 11. Draw a diagram illustrating how you modify Figure 1 to realize quadrature-amplitude-modulation transmission. (7 marks) Assignment 2 (2017 SP2) Page 2 of 4EEET 3028 Communication Systems University of South Australia School of Engineering Digital receiver 12. Draw a block diagram of the receiver. Explain how noise on the communication channel affects the signals in the receiver, and how it affects the symbol estimates and the bit estimates in the receiver. (5 marks) 13. Consider the transmit pulse p(τ) from above, the receive filter with impulse response h(τ) = sinc t −T T ss=2 rect t −T T ss=2 : and transmission of the single modulation symbol x0 = +1 over a noise-free communication channel. Plot the transmit signal and the signal at the receive-filter output with MATLAB. Indicate the optimal sample time, and give reasons. Discuss why the receive filter h(τ) is not optimal; give the expression for the optimal receive filter. (10 marks) 14. Consider now transmission of the bit sequence in question 10 over a noise-free communication channel. An optimal receiver filter is used. Plot in MATLAB the transmit signal and the output of the receive filter. Discuss if there is intersymbol interference (ISI) in this system (10 marks) Decision regions Consider a digital transmission system with the modulation alphabet A4 = f−3; −1; +1; +3g and an AWGN channel. Assume that transmit and receive filter are chosen such that we have zero ISI. Consider the samples of the matched-filter output (discrete-time signal model) zk = ak + wk; where ak 2 A4 and wk is white Gaussian noise with zero mean and variance σw 2 . 15. Sketch by hand the conditional probabilities (likelihoods) p(zkjak) for the modulation symbols ak 2 A4. Indicate the decision regions, and discuss which symbols you expect to experience high or low error rates. (7 marks) 16. Assume that ak = +1 is transmitted, and that σw 2 = 0:2. Compute the error probability. (8 marks) 17. Confirm the error probability result with MATLAB. Repetitively transmit the symbol ak = +1 for 105 times. For each transmission, Assignment 2 (2017 SP2) Page 3 of 4EEET 3028 Communication Systems University of South Australia School of Engineering • Generate a Gaussian noise sample with variance wk and compute zk. • Make a decision following question 15, and detect whether an error occurs. Count the number of errors and compute the error probability. (10 marks) 18. Repeat questions 15 and 16 for ak = +3. Compare and contrast with the case ak = +1. (5 marks) Assignment 2 (2017 SP2) Page 4 of 4